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To represent the sum o f " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 7 " and \+ " }{XPPEDIT 18 0 "1/5" "6#*&\"\"\"F$\"\"&!\"\"" }{TEXT -1 40 " for ex ample, they would simply write " }{XPPEDIT 18 0 "1/3 + 1/5" "6#,&*&\" \"\"F%\"\"$!\"\"F%*&F%F%\"\"&F'F%" }{TEXT -1 125 ". Actually, there' s not a thing wrong with this, although it is tempting for us to go ah ead and 'add' them together to get " }{XPPEDIT 18 0 "8/15" "6#*&\"\") \"\"\"\"#:!\"\"" }{TEXT -1 10 ". But " }{XPPEDIT 18 0 "1/3 + 1/5" "6#,&*&\"\"\"F%\"\"$!\"\"F%*&F%F%\"\"&F'F%" }{TEXT -1 31 " is a perfe ctly good name for " }{XPPEDIT 18 0 "8/15" "6#*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 50 " , a fact we recognize when we write the equation " } {XPPEDIT 18 0 "1/3 + 1/5 = 8/15" "6#/,&*&\"\"\"F&\"\"$!\"\"F&*&F&F&\" \"&F(F&*&\"\")F&\"#:F(" }{TEXT -1 126 " . The problem the Egyptians h ad was that although they had a notation for the unit fractions, i.e. , fractions of the form " }{XPPEDIT 18 0 "1/n " "6#*&\"\"\"F$%\"nG!\" \"" }{TEXT -1 64 ", they did not have a compact notation for the gener al fraction " }{XPPEDIT 18 0 " m/n" "6#*&%\"mG\"\"\"%\"nG!\"\"" } {TEXT -1 144 ". Some might say their numeration system was faulty, bu t that would be overly critical, since they were the first (as far as \+ we know) to have " }{TEXT 262 4 "any " }{TEXT -1 34 " way of giving n ames to fractions." }}{PARA 260 "" 0 "" {TEXT -1 1 "\n" }{TEXT 276 8 " Question" }{TEXT -1 78 ": How do you suppose an Egyptian scribe woul d have written 3/8 ? 3/5 ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 174 "Even though the Egyptian method of writi ng fractions stuck around for a long time, people were still very awar e of its limitations. A good example of this is found in the " } {TEXT 263 8 "Almagast" }{TEXT -1 463 " , written by the Greek Mathemat ician, Geographer, Scientist, etc, Ptolemy in the first century AD. T his book is sometimes referred to as the first trig book because it co ntains tables of chords for the equivalents of the sine and cosine fun ction for use in astronomical calculations. Ptolemy\nexplains that he is using the Babylonian method of writing fractions rather than the E gyptian method because of the embarrassments that the Egyptian method \+ often cause." }}{PARA 0 "" 0 "" {TEXT -1 76 "\nMuch of what we know ab out Egyptian fractions has been inferred from the " }{TEXT 279 13 "R hind papyrus" }{TEXT -1 398 " , written by the scribe Ahmes around 165 0 BC. This book consists mainly of 84 word problems of a diverse natur e, plus a few tables to aid the young scriblets that Ahmes had charge \+ of with their calculations. Generally speaking, Ahmes seemed to be hap py to write the answer to a problem as a whole number plus a sum of un it fractions. He did not use any unit fraction more than once in the \+ answer." }}{PARA 260 "" 0 "" {TEXT -1 1 "\n" }{TEXT 277 7 "Problem" } {TEXT -1 70 " Establish the following algebraic identity: For any \+ n except 0, " }{XPPEDIT 18 0 "1/n = 1/(n+1) + 1/(n(n+1)" "6#/*&\" \"\"F%%\"nG!\"\",&*&F%F%,&F&F%F%F%F'F%*&F%F%-F&6#,&F&F%F%F%F'F%" } {TEXT -1 5 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 144 "With this identity, you can see that a fraction can alwa ys be written in lots of different ways.\nThe largest table in the Rhi nd Papyrus is the " }{XPPEDIT 18 0 "2/n" "6#*&\"\"#\"\"\"%\"nG!\"\"" }{TEXT -1 221 " table, where Ahmes gives decompositions of these frac tions into sums of unit fractions. Most of the entries in the table co me from the decomposition you get by multiplying the above identity on both sides by two. Thus " }{XPPEDIT 18 0 "2/7 = 2/8 + 2/(7*8)" " 6#/*&\"\"#\"\"\"\"\"(!\"\",&*&F%F&\"\")F(F&*&F%F&*&F'F&F+F&F(F&" } {TEXT -1 5 " = " }{XPPEDIT 18 0 "1/4 + 1/28" "6#,&*&\"\"\"F%\"\"%! \"\"F%*&F%F%\"#GF'F%" }{TEXT -1 130 " . \nMaple acts very naturally \+ with Egyptian fractions. It simply adds them up and reduces to lowest terms. Thus if you enter " }{XPPEDIT 18 0 "1/3 + 1/5" "6#,&*&\"\"\" F%\"\"$!\"\"F%*&F%F%\"\"&F'F%" }{TEXT -1 24 " , the output will be \+ " }{XPPEDIT 18 0 " 8/15" "6#*&\"\")\"\"\"\"#:!\"\"" }{TEXT -1 156 ". \+ We have to do something to keep a record of the decomposition, and the simplest way to do that is to make it into a list. Thus an egyptian \+ fraction for " }{XPPEDIT 18 0 "8/15 " "6#*&\"\")\"\"\"\"#:!\"\"" } {TEXT -1 11 " would be \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " ef := [1/3,1/5]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#efG7$#\"\"\"\" \"$#F'\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "\nIn order to convert to the usu al form we could type \n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " con vert(ef,`+`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\")\"#:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Note: the quotes around the + are backquotes ` not forw ard '. " }}{PARA 0 "" 0 "" {TEXT -1 445 "\nOne thing that would be \+ nice to do is take an egyptian fraction and print out an equation whos e left hand side is the usual form of the fraction and whose right han d side is the fraction written as a sum of unit fractions. Because Map le is so intent on writing fractions in the usual form, we must play a trick on it. First we will convert each of the unit fractions in the \+ egyptian fraction to a string. This can be done using the Maple word \+ " }{TEXT 278 3 "map" }{TEXT -1 11 " , like so\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " ef2 := map(convert,ef,symbol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ef2G7$%$1/3G%$1/5G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Now we can write the desi red equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " convert(ef,`+`)=convert(ef2,`+`); " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/#\"\")\"#:,&%$1/3G\"\"\"%$1/5GF)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "We will define some Maple words " }{TEXT 285 4 "fibo" }{TEXT -1 2 ", " }{TEXT 286 9 "symbolize" }{TEXT -1 6 ", and " }{TEXT 287 9 "activate " }{TEXT -1 108 "below which auto mate the process of decomposing fractions into sums of unit fractions \+ and displaying these." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" } {TEXT 282 19 "Fibonnaci's theorem" }{TEXT -1 2 " " }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 207 " The Egyptian system of writing fractions as sums of unit fractions stuck around even after much more efficient systems were developed. Fibonnaci was aware of the system in 1200 AD and incl uded in his book " }{TEXT 280 11 "Liber Abaci" }{TEXT -1 96 " a meth od for writing any fraction as a sum of unit fractions. His method wa s very natural: \n\n" }{TEXT 281 18 "Fibonnaci's method" }{TEXT -1 310 ": Take the fraction you wish to express in the Egyptian manner a nd subtract from it the largest unit fraction which is not larger than it. If the remainder is not itself a unit fraction, then repeat the \+ process on the remainder. Continue this until the remainder is a unit fraction.\n\nFor example, suppose \n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 " a := 4/5 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"%\" \"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The largest unit fraction \+ not larger than it is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\" " }{TEXT -1 15 ". Subtract it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 " a := a - 1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"$\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 " The largest unit fra ction not larger than " }{XPPEDIT 18 0 "3/10" "6#*&\"\"$\"\"\"\"#5!\" \"" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "1/4 " "6#*&\"\"\"F$\"\"%!\"\" " }{TEXT -1 19 ". Subtract that.\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " a := a - 1/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#\"\"\" \"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Fibonnaci's method leads to the decomposition " }{XPPEDIT 18 0 "4 / 5 = 1 / 2 + 1 / 4 + 1 / 20" "6#/*&\"\"%\"\"\"\" \"&!\"\",(*&F&F&\"\"#F(F&*&F&F&F%F(F&*&F&F&\"#?F(F&" }{TEXT -1 4 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 156 "It is not clear whether Fibonnaci ha d a proof that his method always worked, but a proof that it does alwa ys work can be fashioned from the following lemma." }}{PARA 259 "" 0 " " {TEXT 268 5 "Lemma" }{TEXT -1 8 " Let " }{XPPEDIT 18 0 "p/q" "6#* &%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 52 " be any fraction which is not a \+ unit fraction. Let " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" } {TEXT -1 52 " be the largest unit fraction less than or equal to " } {XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 9 " . Then \+ " }{XPPEDIT 18 0 "p/q-1/n" "6#,&*&%\"pG\"\"\"%\"qG!\"\"F&*&F&F&%\"nGF( F(" }{TEXT -1 15 " is a fraction " }{XPPEDIT 18 0 "r/s" "6#*&%\"rG\"\" \"%\"sG!\"\"" }{TEXT -1 14 " with r < p ." }}{PARA 0 "" 0 "" {TEXT 269 6 "Proof " }{TEXT -1 54 " The argument for the lemma is simple. First write " }{XPPEDIT 18 0 " p/q - 1 / n = (n*p - q)/(n*q)" "6#/, &*&%\"pG\"\"\"%\"qG!\"\"F'*&F'F'%\"nGF)F)*&,&*&F+F'F&F'F'F(F)F'*&F+F'F (F'F)" }{TEXT -1 13 " . Now if " }{XPPEDIT 18 0 "p <= n*p-q*%?;" "6 #1%\"pG,&*&%\"nG\"\"\"F$F(F(*&%\"qGF(%#%?GF(!\"\"" }{TEXT -1 15 ", the n adding " }{XPPEDIT 18 0 "q-p" "6#,&%\"qG\"\"\"%\"pG!\"\"" }{TEXT -1 44 "\nto both sides of the inequality gives that " }{XPPEDIT 18 0 " n*p - p >= q" "6#1%\"qG,&*&%\"nG\"\"\"%\"pGF(F(F)!\"\"" }{TEXT -1 15 " . But then " }{XPPEDIT 18 0 "1/( n*p - p) <= 1/q" "6#1*&\"\"\"F%,& *&%\"nGF%%\"pGF%F%F)!\"\"F**&F%F%%\"qGF*" }{TEXT -1 40 ". Now multipl y both sides by p and get " }{XPPEDIT 18 0 "1/(n-1) <= p/q" "6#1*&\"\" \"F%,&%\"nGF%F%!\"\"F(*&%\"pGF%%\"qGF(" }{TEXT -1 10 ". Since " } {XPPEDIT 18 0 " p/q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 69 " is not a unit fraction, but is less than 1, we have that n > 2, and " } {XPPEDIT 18 0 "1/(n-1)" "6#*&\"\"\"F$,&%\"nGF$F$!\"\"F'" }{TEXT -1 32 " is a unit fraction larger than " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$% \"nG!\"\"" }{TEXT -1 23 " which is smaller than " }{XPPEDIT 18 0 "p/q " "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 58 ". This contradicts the cho ice of n, and we conclude that " }{XPPEDIT 18 0 "n*p - q < p" "6#2,&* &%\"nG\"\"\"%\"pGF'F'%\"qG!\"\"F(" }{TEXT -1 3 ". " }{TEXT 270 3 "qed " }{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT 271 8 "Theorem " }{TEXT -1 46 " Fibbonaci's method works for all fractions " }{XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 272 5 "Proof" }{TEXT -1 17 " Starting with " }{XPPEDIT 18 0 "p /q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 17 " and subtracting " } {XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 28 " we get a s maller fraction " }{XPPEDIT 18 0 "(n*p-q)/(n*q\} = p1/q1" "6#/<#*&,&* &%\"nG\"\"\"%\"pGF)F)%\"qG!\"\"F)*&F(F)F+F)F,*&%#p1GF)%#q1GF," }{TEXT -1 14 ". Further, " }{XPPEDIT 18 0 "p1 < p" "6#2%#p1G%\"pG" }{TEXT -1 85 " by the Lemma. If p1 = 1, we are done. If not, then it will \+ be after no more than " }{XPPEDIT 18 0 "p1 - 1" "6#,&%#p1G\"\"\"F%!\" \"" }{TEXT -1 33 " more applications of the Lemma. " }{TEXT 273 3 "qed " }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 189 "With just a little more work, we can modify Fibbonaci's method to get a representation o f any rational number as a sum of distinct unit fractions, each smalle r than a given unit fraction. " }}{PARA 259 "" 0 "" {TEXT 274 7 "Theo rem" }{TEXT -1 40 " Given any (positive) rational number " } {XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 23 " and any unit fraction " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"qG!\"\"" }{TEXT -1 65 " can be written as a sum of distinct unit fractions smaller than \+ " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 2 ".\n" }} {PARA 0 "" 0 "" {TEXT 275 6 "Proof." }{TEXT -1 21 " Since the series " }{XPPEDIT 19 1 "Sum(1/i,i = n+1 .. infinity);" "6#-%$SumG6$*&\"\" \"F'%\"iG!\"\"/F(;,&%\"nGF'F'F'%)infinityG" }{TEXT -1 51 " diverges, there is a positive integer k so that" }{XPPEDIT 19 1 "Sum(1/i,i = n +1 .. n+k) <= p/q" "6#1-%$SumG6$*&\"\"\"F(%\"iG!\"\"/F);,&%\"nGF(F(F(, &F.F(%\"kGF(*&%\"pGF(%\"qGF*" }{TEXT -1 6 " and " }{XPPEDIT 19 1 "p/q < Sum(1/i,i = n+1 .. n+k+1);" "6#2*&%\"pG\"\"\"%\"qG!\"\"-%$SumG6$*&F &F&%\"iGF(/F-;,&%\"nGF&F&F&,(F1F&%\"kGF&F&F&" }{TEXT -1 49 ". If the first inequality is an equality, then " }{XPPEDIT 18 0 "p/q" "6#*&%\" pG\"\"\"%\"qG!\"\"" }{TEXT -1 113 " has been written in the required m anner. If the first inequality is strict, then the remainder is a f raction " }{XPPEDIT 18 0 "r/s" "6#*&%\"rG\"\"\"%\"sG!\"\"" }{TEXT -1 20 " which is less than " }{XPPEDIT 18 0 "1/(n+k+1)" "6#*&\"\"\"F$,(% \"nGF$%\"kGF$F$F$!\"\"" }{TEXT -1 73 ". Consequently, Fibonacci's me thod can be used from here on to express " }{XPPEDIT 18 0 "r/s" "6#*&% \"rG\"\"\"%\"sG!\"\"" }{TEXT -1 101 " as a sum of distinct unit fracti ons which are all distinct from the consecutive ones already used. " }{TEXT 264 3 "qed" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 60 "A Maple word to carry out (the modified) Fibonnaci method. " } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 " Here is the definition of a Map le word " }{TEXT 265 4 "fibo" }{TEXT -1 138 " which implements Fibo nnaci's method for decomposing a fraction into a sum of unit fractions . It uses two Maple words in its definition," }{TEXT 266 6 " numer" } {TEXT -1 49 ", which gives the numerator of its argument, and " } {TEXT 267 5 "trunc" }{TEXT -1 160 ", which returns the integer part of its argument. It includes the modification established above which g uarantees a decomposition into a sum of unit fractions " }{XPPEDIT 18 0 "1/n" "6#*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 15 " with n > m. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "fibo := proc(x,m)\n local \+ z,l,i,n,k;\n k := m+1;\n z := x;\n l \+ := NULL;\n for i while 1 < numer(z) or not l = NULL and z = l[-1]\n do\n n := max(k,trunc(1/z)+1);\n \+ k := k+1;\n z := z-1/n;\n l := l,1/n;\n od;\n l := [l,z];\n end:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 " To get a decomposition of 5/2 3 into unit fractions, enter" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " ef := fibo(5/23,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#efG7%#\"\"\"\"\"&#F'\"#e#F'\"%qm" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "We want to write the fraction as a sum, \+ but simply converting the list to the type `+` doesn't work because Ma ple adds the fractions together." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "convert(ef,`+`);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## \"\"&\"#B" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "We can convert the \+ numerator and denominator of the fraction to the type symbol. Then th e fractions will not combine when they are added. " }{TEXT 284 9 "Sym bolize" }{TEXT -1 68 " takes the output from fibo and outputs a sum of symbolic fractions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "has type(ef,list);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "symbolize := proc(frac,k)\n local \+ i,fracs;\n fracs := frac;\n if not hastype(fracs,list) then \n \+ fracs := fibo(frac,k) fi;\n convert([seq(convert(numer(fracs[i]),sym bol)/\nconvert(denom(fracs[i]),symbol),\ni=1..nops(fracs))],`+`) end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "symbolize(fibo(2/301,1) ,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"1G\"\"\"%$151G!\"\"F&*& F%F&%&45451GF(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "It is necess ary to be able to convert an eqyptian fraction from the inert form to the active form. That's what " }{TEXT 283 9 "activate " }{TEXT -1 5 "does." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 806 "activate := proc (frac)\n local n,m,i,j,k,bill,sam,nu,de,ans;\n ans := 0;\n if hastype( frac,`+`) then\n for k from 1 to nops(frac) do\n n := convert(numer(op (k,frac)),bytes);\n m := convert(denom(op(k,frac)),bytes);\n bill := [ seq(48,i=1..nops(n))];\n sam := [seq(48,i=1..nops(m))];\n n := n - bil l;\n nu := convert([seq(n[nops(bill)-i]*10^i,i=0..nops(bill)-1)],`+`); \n m := m - sam;\n de := convert([seq(m[nops(sam)-i]*10^i,i=0..nops(sa m)-1)],`+`);\n ans := ans+nu/de;\n od;\n else\n n := convert(numer( fr ac ),bytes);\n m := convert(denom( frac ),bytes);\n bill := [seq(48,i= 1..nops(n))];\n sam := [seq(48,i=1..nops(m))];\n n := n - bill;\n nu : = convert([seq(n[nops(bill)-i]*10^i,i=0..nops(bill)-1)],`+`);\n m := m - sam;\n de := convert([seq(m[nops(sam)-i]*10^i,i=0..nops(sam)-1)],`+ `);\n ans := ans+nu/de;\n fi;\n ans;\n end:\n \n " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "activate(symbolize(2/1));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "`&ea` := (fr1,fr2) -> symbol ize(activate(fr1)+activate(fr2),1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$&eaGf*6$%$fr1G%$fr2G6\"6$%)operatorG%&arrowGF)-%*symbolizeG6$,&-%) activateG6#9$\"\"\"-F26#9%F5F5F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f1 := symbolize(1/6,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&%\"1G\"\"\"%\"6G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f2 := symbolize(1/5,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G*&%\"1G\"\"\"%\"5G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "f3 :=f1 &ea f2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #f3G,&*&%\"1G\"\"\"%\"3G!\"\"F(*&F'F(%#30GF*F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "activate(f3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#6\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "b:=fibo(9/1 0,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG7%#\"\"\"\"\"##F'\"\"$# F'\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "symbolize(b,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"1G\"\"\"%\"2G!\"\"F&*&F%F&%\" 3GF(F&*&F%F&%#15GF(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " \+ symbolize(13/10,1) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&%\"1G\"\" \"%\"2G!\"\"F&*&F%F&%\"3GF(F&*&F%F&%\"4GF(F&*&F%F&%\"5GF(F&*&F%F&%#60G F(F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 41 " Investigate these problems. Use fibo ." }} {EXCHG {PARA 260 "" 0 "" {TEXT -1 92 "1.Print out decompositions of th e fractions with an odd numerator and a denominator of 100.\n" }} {PARA 260 "" 0 "" {TEXT -1 81 "2. Find a fraction which fibo decomp oses into the sum of six unit fractions.\n" }}{PARA 260 "" 0 "" {TEXT -1 200 "3. Find a fraction which fibo decomposes into the \+ sum of eight unit fractions.\n\n4. Show that there are infinitely ma ny ways to write a given fraction as a sum of unit fractions. \+ " }}}}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "table of contents" 1 "mathist.mws" "" }{TEXT -1 1 " " }}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }